Duality (projective Geometry) - Duality As A Mapping

Duality As A Mapping

A (plane) duality is a map from a projective plane C = (P,L,I) to its dual plane C* = (L,P,I*) (see above) which preserves incidence. That is, a (plane) duality σ will map points to lines and lines to points (Pσ = L and Lσ = P) in such a way that if a point Q is on a line m ( denoted by Q I m) then Qσ I* mσ ⇔ mσ I Qσ. A (plane) duality which is an isomorphism is called a correlation. The existence of a correlation means that the projective plane C is self-dual.

In the special case that the projective plane is of the PG(2,K) type, with K a division ring, a duality is called a reciprocity. These planes are always self-dual. By the Fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography. If the automorphism involved is the identity, then the reciprocity is called a projective correlation.

A correlation of order two (an involution) is called a polarity. If a correlation φ is not a polarity then φ2 is a nontrivial collineation.

This duality mapping concept can also be extended to higher dimensional spaces so the modifier "(plane)" can be dropped in those situations.